How To Find The Area Of A Rhombus Without Diagonals

Area of Rhombus

Expanse of rhomb is defined as the corporeality of space enclosed or encompassed by a rhomb in a 2-dimensional airplane. Rhombus is a special type of parallelogram, having all sides equal to each other. A rhombus tin can be differentiated from a square through the measure out of its internal angle. The internal angle of a rhombus demand not be a right-angle specifically. The area of a rhombus can be calculated in different ways, depending on the parameters known to us.

1.	What is Expanse of Rhombus?
2.	Expanse of a Rhombus Formula
3.	How to Calculate the Area of a Rhombus?
4.	FAQs on Area of Rhomb

What is Area of a Rhombus?

The area of a rhomb tin exist divers as the amount of space enclosed by a rhomb in a two-dimensional space. It depicts the total number of unit squares that can fit into information technology and it is measured in square units (like cm², thou², in², etc). Rhomb is a parallelogram with the opposite sides parallel, the reverse angles equal, and the adjacent angles supplementary. The following backdrop are used to define a rhomb.

A rhombus is an equilateral quadrilateral because all the sides have equal lengths.
In a rhombus, diagonals bifurcate each other at right angles.
The diagonals are angle bisectors.
The area of a rhombus tin be found in different ways: using base and top, using diagonals, and using trigonometry.

Expanse of a Rhomb Formula

Different formulas can be used to calculate the expanse of a rhombus depending upon the parameters known to the states. The different formulas followed for the calculation of the surface area of a rhombus are,

Using base and height
Using diagonals
Using trigonometry

Formula for Area of Rhombus When Base of operations and Height Are Known

Rhombus is a parallelogram. Nosotros know that the surface area of a parallelogram is given by multiplying base and pinnacle sq units. The same is applied to the rhombus as well.
Surface area of a Rhombus = base × summit sq units

Example: Find the area of a rhomb having a side length of 7 inches and the summit of the rhomb of 10 inches.
Solution: As we know, Surface area = base × pinnacle units²
⇒ Area = vii × ten inchesⁱⁱ
⇒ Area = lxx inches²

Formula for Surface area of Rhombus When Diagonals Are Known

The surface area of a rhombus is equal to one-half the production of the lengths of the diagonals. The formula to calculate the expanse of a rhombus using diagonals is given as,

Area = (d₁ × d₂)/2 sq. units, where, d₁ and d_ii are the diagonals of the rhombus.

Area of rhombus given diagonals

Consider the Rhomb ABCD. Let E be the point of intersection of 2 diagonals. We make the following observations,

Four sides are coinciding.
The diagonals bisect each other.
Four interior angles with opposite angles equal. ⇒ ∠ A = ∠C and ∠B = ∠D
The ii diagonals are Air-conditioning and BD.

The area of the rhombus ABCD = Expanse of ∆ ADC + Area of ∆ ABC

Surface area of the rhombus = 2 × Surface area of ∆ ABC ---(1) (∵ ∆ ABC congruent to ∆ ADC)

Surface area of ∆ ABC

= 1/2 × Base × Pinnacle
= 1/2 × AC × Exist
= i/2 × Air-conditioning × 1/2 × BD (∵Be = BD/2)
= one/4 (AC × BD) --- (2)

Surface area of the Rhombus ABCD

Area = 2 × 1/4 × Air conditioning × BD = 1/2 × AC × BD (From (1) and (2))
⇒ Area = i/2 × diagonal ane × diagonal 2

∴ Area of a Rhomb = 1/2 × diagonal 1 × diagonal 2 units²

Formula for Area of Rhomb When Side and Angles Are Known

Nosotros utilise the concept of trigonometry while computing the surface area when sides and angles are known. We can utilize whatever bending because either the angles are equal or they are supplementary, and supplementary angles have the same sine. Expanse of a rhomb using side and angle is given as,

Area of a Rhombus = side²× sin(A) sq. units, where 'A' is an interior angle.

area of the rhombus using trigonometry

Example: What is the expanse of a rhombus if the length of its side is 4 yards and one of its angle A is 30º.
Solution: As we know, expanse of rhombus = s²× sin(30º)
Area of rhombus = s²× sin(30º) = 4ⁱⁱ × 1/ii
⇒ Area of rhombus = 16 × one/2 = 8 sq yards

How to Summate the Area of a Rhomb?

The dissimilar methods to calculate the area of a rhombus are explained below. There are iii methods for calculating the area of a rhombus, given as:

Method one: Using Base and Pinnacle
Method 2: Using Diagonals
Method 3: Using Trigonometry

Area of Rhombus Using Base and Height

Pace 1: Find and note the base and the acme of the given rhombus. The base of operations is one of its sides of a rhombus, while the summit is the perpendicular distance from the chosen base to the opposite side.
Step 2: Multiply the base and meridian.

The resultant value will give the area of a rhomb.

Area of Rhombus Using Diagonals

Consider a rhombus ABCD, having ii diagonals, i.east. AC & BD.

Step 1: Find the length of both the diagonals, diagonal 1 and diagonal 2.
Footstep 2: Multiply both the lengths, d1, and d2.
Step three: Split up the effect by 2.

The resultant value will give the expanse of a rhombus ABCD.

Area of Rhombus Using Trigonometry

Step 1: Foursquare the length of whatever of the sides.
Step 2: Multiply it past the sine of whatsoever one of the angles.

The resultant value will give the area of a rhombus.

Instance: Consider the rhombus ABCD. AB, BC, CD, DA are the coinciding (equal) sides. AC and BD are the diagonals, and they see at E. Given CD = 17 feet and AE = 8 anxiety. Find the expanse of ABCD.

At present nosotros know diagonal ane, Air conditioning = 16 feet.

Next, we need to calculate BD.
BD = BE + ED = 2 × Exist

We all the same have an unknown, BE.
Pythagorean theorem states that

BC^two = BE² + EC^two
BC = 17 anxiety (∵ CD =BC as all the sides are congruent)
EC = 8 feet (∵ AE = EC as the diagonals bifurcate each other)
17^two = Exist^two+ viii²
⇒ BE² = 289 - 64
= 225

∴ BE = fifteen feet and BD = thirty feet

Lets substitute all the values in the expanse of the rhombus formula.

Surface area of the Rhombus = 1/ii × d₁ × d₂ sq units
= 1/2 × BD × Air-conditioning sq feet
= 1/ii × 30 × 16 sq anxiety
⇒ Area of rhombus = 240 sq feet

Tips and Tricks:

Retrieve that the height is non the same every bit the length of the side of a rhombus.
The expanse of a rhombus can be found in iii ways: when diagonals are given, when an angle and a side are given, when angle and superlative are given.
Utilise the Pythagorean theorem to find the 2nd diagonal if the measures of one diagonal and side are given.

Examples on Area of Rhombus

Example 1: Using the area of a rhomb formula, observe the area of the rhombus given in the figure below.

Solution:

Area of a rhombus= 1/ii × BD × AC

BD = ii × Be
= 2 × 8
= sixteen yards

AC = ii × AE = two × ten = 2 yards
⇒ Area = 1/2 × 16 × 20
= 8 × 20
= 160 yards^two

Answer: Expanse of the rhombus = 160 yards²
Example 2: The sides of a rhomb ABCD measure 5 inches, while the length of its one diagonal Ac is 8 in. Calculate its area.

Solution:

Given a side and a diagonal, nosotros will find the other diagonal. Allow O exist the point of intersection of diagonals.

Surface area = (Air-conditioning × BD)/ii sq inches
⇒ AC = 8 in
⇒ AO = 4 in (∵ AO = ane/2 Air conditioning)

To find the other diagonal BD, consider AOD.

By Pythagorean theorem, AD^two= AO^two + ODⁱⁱ
⇒ 25² = 4^two + OD^two
⇒ OD² = 25 −16
⇒ OD² = 9
⇒ OD = three in
⇒ BD = 6 in (∵BD = 2 × OD)
Expanse = (8 × 6) ÷ two sq inches
Expanse = 24 sq inches

Answer: Area of the rhomb = 24 sq inches
Example 3: The area of a rhomb is given equally 256 foursquare units. If the length of ane of the diagonals is 8 units, find the length of its other diagonal.

Solution:

We know surface area of a rhombus tin can be calculated using diagonals as,

Area = (i/2) × (diagonal₁ × diagonal₂)

⇒ 256 = (1/2) × (8 × diagonal₂)
diagonal₂ = (256 × two)/8
diagonal₂ = 64

Answer: Length of the other diagonal of the given rhombus = 64 units

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Exercise Questions on Surface area of Rhombus

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FAQs on Area of Rhombus

What is the Area of Rhombus?

The area of a rhombus is the total corporeality of space enclosed or encompassed by a rhomb in a 2-dimensional plane. It is expressed in square units(like cm², mⁱⁱ, in^two, etc).

What is the Formula of Finding the Area of a Rhombus?

Different formulas can be used to summate the expanse of a rhomb depending upon the parameters known to us. Using base and altitude the formula is given as, Area of a Rhombus = base × top sq units. Area of rhombus using diagonals is: Area = (d_one × d_two)/two sq. units, where d_i and d₂ are the diagonals of the rhomb. Applying the concept of trigonometry using side and angle, we tin can follow the formula: Surface area of a Rhombus = side²× sin(A) sq. units, where 'a' is an interior angle.

How practise you Discover the Side of a Rhombus With the Diagonals?

The area of a rhombus can exist calculated using the lengths of diagonals. The formula to find area in this case is given equally: Area = (d₁ × d_two)/2 sq. units, where d₁ and d₂ are the diagonals of the rhomb.

Are the Areas of Rhombus and Square Equal?

No, the area of the rhombus and square are not equal. However, their area could be calculated the same way, given their measures. Expanse of a rhombus or any parallelogram = base of operations × tiptop. In a rhombus, the side and height are not the same. Still, the area of the square = side×side, wherein the side could also be the height of the square. A foursquare is a rhomb because information technology has iv sides, and each side is the same length. Still, the square is further defined equally a shape that has iv equal angles of 90 degrees. Therefore, a square is a rhombus. Yet, a rhomb is not necessarily a square. And then their areas cannot be the same.

How to Notice Area of Rhombus When Side and Altitude are Given?

The area of a rhombus can exist calculated when the length of the base of operations or side and the height is given. Here, superlative refers to the perpendicular altitude betwixt the parallel sides, one of which we took equally a base of operations. The formula to find an expanse, in this instance, is given every bit area of a rhomb = base of operations × height sq units.

What is the Difference Between Area of Rhomb and Square?

The area of a foursquare is square of its side i.e., due south × s, where 's' is the length of the side of a square whereas the area of a rhomb is ½ × d1 × d2, where d1 and d2 are the lengths of the diagonals.

What is the Difference Between Area of Rhombus and Rectangle?

Area of a rectangle is l × b, where 'l' is the length of the rectangle and 'b' is the breath of the rectangle whereas the expanse of a rhomb is ½ × d1 × d2, where d1 and d2 are the lengths of the diagonals.

What is the Divergence Between Area of Rhomb and its Perimeter?

The perimeter of a rhombus is the total measure of its boundary and it is calculated by adding the length of all its sides whereas area of a rhombus or whatsoever parallelograms is a product of its base of operations and height, i.e., base × elevation. Hence the area of rhombus is one-half the product of ts diagonals given as ½ × d1 × d2, where d1 and d2 are the lengths of the diagonals.

What is the Use of Surface area of a Rhombus Calculator?

Area of a rhombus calculator is an online tool with the assist of which we can easily evaluate the area of rhomb in seconds. To determine the value nosotros demand to enter the specific parameters such every bit value of the diagonals. Effort Cuemath'due south online rhombus area calculator for quick calculations and solve your issues related to surface area of rhombus within seconds.

What is the Distance if Area of Rhomb is Given?

To summate the altitude or meridian when the area is given, nosotros require the length of the base. The formula that can be applied to calculate height is given as Expanse/base units.